4The Concept of CorrelationAssociation or relationship between two variablesCo-relate?rrelationXYCovary---Go together

5Patterns of CovariationZero or no correlationXYCorrelationCovaryGo togetherXYXYNegative correlationPositive correlation

6Scatter plots allow us to visualize the relationshipsThe chief purpose of the scatter diagram is to study the nature of the relationship between two variablesLinear/curvilinear relationshipDirection of relationshipMagnitude (size) of relationshipScatter Plots

13The Measurement of CorrelationThe Correlation CoefficientThe degree of correlation between two variables can be described by such terms as “strong,” ”low,” ”positive,” or “moderate,” but these terms are not very precise.If a correlation coefficient is computed between two sets of scores, the relationship can be described more accurately.A statistical summary of the degree and direction of relationship or association between two variables can be computed

15The Pearson Product-Moment Correlation CoefficientRecall that the formula for a variance is:If we replaced the second X that was squared with a second variable, Y, it would be:This is called a co-variance and is an index of the relationship between X and Y.

16Conceptual Formula for Pearson rThis formula may be rewritten to reflect the actual method of calculation

17Calculation of Pearson rYou should notice that this formula is merely the sum of squares for covariance divided by the square root of the product of the sum of squares for X and Y

18Formulae for Sums of SquaresTherefore, the formula for calculating r may be rewritten as:

20An ExampleSuppose that a college statistics professor is interested in how the number of hours that a student spends studying is related to how many errors students make on the mid-term examination. To determine the relationship the professor collects the following data:

23Calculating the Correlation Coefficient= -82 / √(56)(162.1)=Thus, the correlation between hours studied and errors made on the mid-term examination is -0.86; indicating that more time spend studying is related to fewer errors on the mid-term examination. Hopefully an obvious, but now a statistical conclusion!

26The Pearson r and Marginal DistributionThe marginal distribution of X is simply the distribution of the X’s; the marginal distribution of Y is the frequency distribution of the Y’s.YBivariate relationshipBivariate Normal DistributionX

27Marginal distribution of X and Y are precisely the same shape.Y variableX variable

28Interpreting r, the Correlation CoefficientRecall that r includes two types of information:The direction of the relationship (+ or -)The magnitude of the relationship (0 to 1)However, there is a more precise way to use the correlation coefficient, r, to interpret the magnitude of a relationship. That is, the square of the correlation coefficient or r2.The square of r tells us what proportion of the variance of Y can be explained by X or vice versa.

29How does correlation explain variance?Suppose you wish to estimate Y for a given value of X.highHow does correlation explain variance?ExplainedVariable YFree to Vary49% of variance is explainedExplainedlowlowhighVariable XAn illustration of how the squared correlation accounts for variance in X, r = .7, r2 = .49

30Now, let’s look at some correlation coefficients and their corresponding scatter plots.